The structure of \(\Pi\) spaces. I
type: Item
instance of: scholarly article
title: The structure of \(\Pi\) spaces. I
zbMATH Open document ID: 0548.46021
author: Shaozong Yan
published in: Chinese Annals of Mathematics. Series B
publication date: 1984-01-01T00:00:00Z
review text: Let \(\Pi\) denote a Krein space with an inner product \((.,.)\). A decomposition \(\Pi =H_+\oplus H_-\) is called a regular decomposition if \(H_{\pm}\) are Hilbert spaces relative to the inner products \((.,.)\). Let \(A\) denote a linear operator from the Hilbert space \(H_-\) to \(H_+\). Let \(L_ A=\{x+Ax,\) \(x\in D(A)\equiv\) domain of \(A\)\}. The author here investigates various properties of \(A\). He then uses these ideas to study the direct sum decomposition of \(\Pi\)-spaces and corresponding projections. In particular, the author gives a new proof of the following: Theorem: A linear subspace \(L\) of \(\Pi\) is a complete subspace (i.e. a \(\Pi\)-type space and closed) iff \(\pi =L\oplus L^{\perp}\).
Mathematics Subject Classification ID: 46C20
zbMATH DE Number: 3873986
zbMATH Keywords: Pontryagin space
zbMATH Keywords: \(\Pi_ k\)-space
zbMATH Keywords: Krein space
zbMATH Keywords: regular decomposition
zbMATH Keywords: direct sum decomposition
zbMATH Keywords: complete subspace
reviewed by: T. K. Mukherjee
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3873986